Due to the significant variations in unmanned aerial vehicle (UAV) altitude and horizontal mobility, it becomes difficult for any single network to ensure continuous and reliable threedimensional coverage. Towards that end, the space-air-ground integrated network (SAGIN) has emerged as an essential architecture for enabling ubiquitous UAV connectivity. To address the pronounced disparities in coverage and signal characteristics across heterogeneous networks, this paper formulates UAV mobility management in SAGIN as a constrained multi-objective joint optimization problem. The formulation couples discrete link selection with continuous trajectory optimization. Building on this, we propose a two-level multi-agent hierarchical deep reinforcement learning (HDRL) framework that decomposes the problem into two alternately solvable subproblems. To map complex link selection decisions into a compact discrete action space, we conceive a double deep Q-network (DDQN) algorithm in the top-level, which achieves stable and high-quality policy learning through double Q-value estimation. To handle the continuous trajectory action space while satisfying quality of service (QoS) constraints, we integrate the maximum-entropy mechanism of the soft actor-critic (SAC) and employ a Lagrangian-based constrained SAC (CSAC) algorithm in the lower-level that dynamically adjusts the Lagrange multipliers to balance constraint satisfaction and policy optimization. Moreover, the proposed algorithm can be extended to multi-UAV scenarios under the centralized training and decentralized execution (CTDE) paradigm, which enables more generalizable policies. Simulation results demonstrate that the proposed scheme substantially outperforms existing benchmarks in throughput, link switching frequency and QoS satisfaction.